In this paper, we study some properties of core-EP inverse of square matrices. Firstly, we extend the obtained theorem proved by K.M. Prasad and K.S. Mohana. Then some properties of core-EP inverse have been given, through applying the conditions (AX) * = AX, XA k+1 = A k and R(X) = R(X *) = R(A k). Secondly, we get some characterizations of core-EP inverse by employing the conditions AX = P A k and XA = P R(A k),N((A k+1) * A). Finally, we get some properties of core-EP inverse by utilizing the condition A k+1 X = A k P A k,